Hi,
I'm not sure if I'm knowledgeable enough to give advice on Shanghai. I could do a much better job with Shenzhen

I haven't been to many places outside of the areas that the guy has already mentioned (Pudong, Bund, French concession) but I can try.

Personally apart from those places I would definitely recommend Xintiandi (although it's kinda within to the French concession anyway ) for a cool (and not cheap) European al fresco dining experience. I know some would ask why would you try European stuff in China but I guess good stuff is good be it Europe, China or anything else and it'a a really great destination.

Jewish ghetto. To be honest nothing that much exciting there from a visual point of view (just some old buildings) but reading some history on it and then seeing it in person is really exciting. There are guided tours taking place by a Jewish guy named Dvir Bar-Gal who knows Shanghai Jewish history better than anyone else and it should be very useful for those who are interested in Jewish heritage and history.

Buildings by a Slovakian-Hungarian architect Laszlo Hudec. He largely shaped the modern face of Shanghai in the 1930's. There are many buildings still remaining which were designed by him back in those days but the most iconic one is Park Hotel (right in the People's Square) which remains one of the most recognizable landmarks in Shanghai. And perhaps the Union Brewery in Yi Chang Road.

Waterhouse Hotel and surrounding area at South Bund which has a number of beautifully restored former Art Deco style warehouses and industrial buildings.

Moganshan Lu and M50 with beautiful street art and art galleries is definitely a must see too. It's near to Union Brewery so both can be done at the same time.

Most likely, you'll take the maglev out of Pudong Airport, but visiting the Hongqiao Transportation hub at the other end of the city is equally impressive (if not more). It's a combination domestic airport, high speed train terminal, inter-city bus depot, and eventually, the Maglev and several metro lines are supposed to go there. It's one of the most ambitious inter-modal hubs I've seen. It was a bit of an effort to get to when I was there in 2010, though

if you're staying where you're staying near the people's square, be sure to wander over to huanghe to score some dumplings from both yang's fried and jia jia tang bao - the two best places for xiao long bao probably on earth (at least, based on my extensive travels). also, if you wake up early, walking the bund in the early morning (sunrise-ish) is a brilliant, never-forget-for-the-rest-of-your-life moment.

if you're staying where you're staying near the people's square, be sure to wander over to huanghe to score some dumplings from both yang's fried and jia jia tang bao - the two best places for xiao long bao probably on earth (at least, based on my extensive travels). also, if you wake up early, walking the bund in the early morning (sunrise-ish) is a brilliant, never-forget-for-the-rest-of-your-life moment.